In most Bayesian analyses, the posterior density \(\pi (\theta \vert \mathcal {D})\)  is analytically intractable and practitioners must instead rely on approximate posterior computations.
There are myriad approximate inference algorithms (e.g. Laplace approximation, Hamiltonian Monte Carlo, coordinate ascent mean-field variational inference, to name just a few).
All of these algorithms aim to approximate the same posterior \(\Pi (\theta \vert \mathcal {D}).\) 
Log predictive-density is often used to compare the quality of different approximations, with higher \(\textrm {lpd}\)  values assumed to reflect more accurate approximations, in the context of variational inference [1]}, [2]}, [3]}, [4]}, [5]} and Bayesian deep learning [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}.
